Departure — Page 266, Lesson 237

I want to walk you through a really useful technique in departure calculations — how to find departure at one latitude when you already know it at another. This comes up when an aircraft flies a series of rhumb line tracks and you need to work out the final position relative to the start. Let's start with the example given. An aircraft leaves position 'G' at latitude 40° South and flies three rhumb line legs: G to H on a true track of 180° for 240 nautical miles, then H to J on a true track of 270° for 240 nautical miles, then J to K on a true track of 000° for 240 nautical miles. The question asks: what is the rhumb line bearing and distance of K from G? Now, the key insight here is that the first leg — 180° true, due south — is a change of latitude only. Since 1 minute of latitude equals 1 nautical mile along a meridian, 240 nautical miles south means a change of latitude of 4 degrees. So H is at latitude 44° South. The second leg, H to J, is due west along the parallel of latitude 44° South. That's a departure calculation. The basic departure formula is: departure in nautical miles equals change of longitude in minutes multiplied by the cosine of the latitude. So we have 240 equals change of longitude times cosine 44°. Rearranging, change of longitude H to J equals 240 divided by cosine 44°, which works out to 333.6 minutes of longitude. The third leg, J to K, is due north — 000° true — for 240 nautical miles. That brings us back up 4 degrees of latitude, so K ends up at latitude 40° South, the same as G. And the change of longitude from G to K is the same as from H to J: 333.6 minutes. Now we need the departure from G to K, which is along latitude 40° South. Using the basic formula again: departure equals change of longitude in minutes times cosine of the latitude. So 333.6 times cosine 40° gives 255.6 nautical miles. Since the track from G to K is due west, the rhumb line bearing is 270° true. So the answer is 270° true, 255.6 nautical miles. That method works, but there's a faster way. The book gives us a formula that relates departure at two different latitudes: departure at latitude A divided by cosine A equals departure at latitude B divided by cosine B. Rearranged, departure at latitude A equals departure at latitude B multiplied by cosine A over cosine B. In our example, we know the departure at latitude B — that's 240 nautical miles at 44° South. We want departure at latitude A, which is 40° South. So we plug in: departure at 40° equals 240 times cosine 40° divided by cosine 44°. That gives the same 255.6 nautical miles, but in one step instead of two. You don't have to calculate the change of longitude first. Let me be clear about what each term means. Departure is the east-west distance along a parallel of latitude, measured in nautical miles. Cosine of latitude is the trigonometric factor that accounts for the convergence of meridians — at the equator, cosine 0° is 1, so departure equals change of longitude in minutes; at the poles, cosine 90° is 0, so departure is zero regardless of longitude change. That's why the formula works: departure varies directly with the cosine of the latitude. So the key takeaway: when you know departure at one latitude and need it at another, use the ratio of cosines. It saves time and reduces the risk of arithmetic errors. The example we just worked through shows both methods so you can see the advantage.